code: def rydberg_interaction(
      register: list,
      wires=None,
      interaction_coeff: float = 862690,
      max_distance: float = np.inf,
  ):
      r"""Returns a :class:`ParametrizedHamiltonian` representing the interaction of an ensemble of
      Rydberg atoms due to the Rydberg blockade
  
      .. math::
  
          \sum_{i<j} V_{ij} n_i n_j
  
      where :math:`n_i` corresponds to the projector on the Rydberg state of the atom :math:`i`, and
      :math:`V_{ij}` is the van der Waals potential:
  
      .. math::
  
          V_{ij} = \frac{C_6}{R_{ij}^6}
  
      where :math:`R_{ij}` is the distance between the atoms :math:`i` and :math:`j`, and :math:`C_6`
      is the Rydberg interaction constant, which defaults to :math:`862690 \times 2 \pi \text{MHz } \mu \text{m}^6`.
      The unit of time for the evolution of this Rydberg interaction term is in :math:`\mu \text{s}`.
      This interaction term can be combined with laser drive terms (:func:`~.rydberg_drive`) to create
      a Hamiltonian describing a driven Rydberg atom system.
  
      .. seealso::
  
          :func:`~.rydberg_drive`
  
      Args:
          register (list): list of coordinates of the Rydberg atoms (in micrometers)
          wires (list): List of wires containing the wire values for all the atoms. This list should
              have the same length as ``register``. If ``None``, each atom's wire value will
              correspond to its index in the ``register`` list.
          interaction_coeff (float): Defaults to :math:`862690 \times 2 \pi \text{MHz } \mu\text{m}^6`.
              The value will be multiplied by :math:`2 \pi` internally to convert to angular frequency,
              such that only the value in standard frequency (i.e., 862690 in the default example) should be
              passed.
          max_distance (float): Threshold for distance in :math:`\mu\text{m}` between two Rydberg atoms beyond which their
              contribution to the interaction term is removed from the Hamiltonian.
  
      Returns:
          :class:`~.ParametrizedHamiltonian`: a :class:`~.ParametrizedHamiltonian` representing the atom interaction
  
      **Example**
  
      We create a Hamiltonian describing the van der Waals interaction among 9 Rydberg atoms in a square lattice:
  
      .. code-block:: python
  
          atom_coordinates = [[0, 0], [0, 5], [0, 10], [5, 0], [5, 5], [5, 10], [10, 0], [10, 5], [10, 10]]
          wires = [1, 5, 0, 2, 4, 3, 8, 6, 7]
          H_i = qml.pulse.rydberg_interaction(atom_coordinates, wires=wires)
  
      >>> H_i
      HardwareHamiltonian: terms=36
  
      As expected, we have :math:`\frac{N(N-1)}{2} = 36` terms for N=9 atoms.
  
      The interaction term is dependent only on the number and positions of the Rydberg atoms. We can execute this
      pulse program, which corresponds to all driving laser fields being turned off and therefore has no trainable
      parameters. To add a driving laser field, see :func:`~.rydberg_drive`.
  
      .. code-block:: python
  
          dev = qml.device("default.qubit.jax", wires=9)
  
          @qml.qnode(dev, interface="jax")
          def circuit():
              qml.evolve(H_i)([], t=[0, 10])
              return qml.expval(qml.PauliZ(0))
  
      >>> circuit()
      Array(1., dtype=float32)
      """
      if wires is None:
          wires = list(range(len(register)))
      elif len(wires) != len(register):
          raise ValueError(
              "The length of the wires and the register must match."
          )
  
      coeffs = []
      observables = []
      for idx, (pos1, wire1) in enumerate(
          zip(register[:-1], wires[:-1])
      ):
          for pos2, wire2 in zip(
              register[(idx + 1) :], wires[(idx + 1) :]
          ):
              atom_distance = np.linalg.norm(
                  qml.math.array(pos2) - pos1
              )
              if atom_distance > max_distance:
                  continue
              # factor 2pi converts interaction coefficient from standard to angular frequency
              Vij = (
                  2
                  * np.pi
                  * interaction_coeff
                  / (abs(atom_distance) ** 6)
              )  # van der Waals potential
              coeffs.append(Vij)
              with qml.QueuingManager.stop_recording():
                  observables.append(
                      qml.prod(
                          qml.Projector([1], wire1),
                          qml.Projector([1], wire2),
                      )
                  )
                  # Rydberg projectors
  
      settings = RydbergSettings(register, interaction_coeff)
  
      return HardwareHamiltonian(
          coeffs,
          observables,
          settings=settings,
          reorder_fn=_reorder_parameters,
      )
  
file_path: pennylane/pennylane/pulse/rydberg.py
imports: from dataclasses import dataclass
  import numpy
  import pennylane
  from pennylane.pulse import HardwareHamiltonian, HardwarePulse, drive
  from pennylane.wires import Wires
  from pennylane.pulse.hardware_hamiltonian import _reorder_parameters
label_desc: Use image node_id 1 for calling a global function with example usage: rydberg_interaction(register, wires, interaction_coeff, max_distance) and returns: HardwareHamiltonian